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OverviewThis volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton-Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault-Kostant complex, and their applications, A discussion of the relation between geometric quantization and the Marsden Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. The text is suitable for graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics. Full Product DetailsAuthor: Mircea PutaPublisher: Springer Imprint: Springer Edition: 1993 ed. Volume: 260 Dimensions: Width: 15.50cm , Height: 1.70cm , Length: 23.50cm Weight: 1.300kg ISBN: 9780792323068ISBN 10: 0792323068 Pages: 280 Publication Date: 30 June 1993 Audience: College/higher education , Professional and scholarly , Postgraduate, Research & Scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of Contents1 Symplectic Geometry.- 1.1 Symplectic Algebra.- 1.2 Symplectic Geometry.- 1.3 Darboux’s Theorem.- 1.4 Symplectic Reduction.- 1.5 Problems and Solutions.- 2 Hamiltonian Mechanics.- 2.1 Hamiltonian Mechanical Systems.- 2.2 Poisson Bracket.- 2.3 Infinite Dimensional Hamiltonian Mechanical Systems.- 2.4 Problems and Solutions.- 3 Lie Groups. Momentum Mappings. Reduction.- 3.1 Lie Groups.- 3.2 Actions of Lie Groups.- 3.3 The Momentum Mapping.- 3.4 Reduction of Symplectic Manifolds.- 3.5 Problems and Solutions.- 4 Hamilton-Poisson Mechanics.- 4.1 Poisson Geometry.- 4.2 The Lie-Poisson Structure.- 4.3 Hamilton-Poisson Mechanical Systems.- 4.4 Reduction of Poisson Manifolds.- 4.5 Problems and Solutions.- 5 Hamiltonian Mechanical Systems and Stability.- 5.1 The Meaning of Stability.- 5.2 Hamilton’s Equations and Stability.- 5.3 The Energy-Casimir Method.- 5.4 Problems and Solutions.- 6 Geometric Prequantization.- 6.1 Full Quantization and Dirac Problem.- 6.2 Complex Bundles and the Dirac Problem.- 6.3 Geometric Prequantization.- 6.4 Problems and Solutions.- 7 Geometric Quantization.- 7.1 Polarizations and the First Attempts to Quantization.- 7.2 Half-Forms Correction of Geometric Quantization.- 7.3 The Non-Existence Problem.- 7.4 Problems and Solutions.- 8 Foliated Cohomology and Geometric Quantization.- 8.1 Real Foliations and Differential Forms.- 8.2 Complex Foliations and Differential Forms.- 8.3 Complex Elliptic Foliations and Spectral Geometry.- 8.4 Cohomological Correction of Geometric Quantization.- 8.5 Problems and Solutions.- 9 Symplectic Reduction. Geometric Quantization. Constrained Mechanical Systems.- 9.1 Symplectic Reduction and Geometric Prequantization.- 9.2 Symplectic Reduction and Geometric Quantization.- 9.3 Applications to Constrained MechanicalSystems.- 9.4 Problems and Solutions.- 10 Poisson Manifolds and Geometric Prequantization.- 10.1 Groupoids.- 10.2 Symplectic Groupoids.- 10.3 Geometric Prequantization of Poisson Manifolds.- 10.4 Problems and Solutions.- References.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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